JEE Main 2019 (10 January)

LEVELJEE Main

Animated Solution for Mathematics - Conic Sections: The length of the chord of the parabola $x^{2} = 4 y$ having equation $x - 2 y + 42 = 0$ is :

Select Answer:

Visualized Solution

Visualizing the Geometry

Given Parabola: $x^{2} = 4 y$
Given Line: $x - 2 y + 42 = 0$
The line intersects the parabola at two points, forming a chord.

Intersection Strategy

To find the intersection points, we solve the equations simultaneously.
From the parabola, we can express $y$ in terms of $x$ : $y = \frac{x ^{2}}{4}$
We will substitute this into the line's equation.

Substituting $y$

Substitute $y = \frac{x ^{2}}{4}$ into $x - 2 y + 42 = 0$ :
$x - 2 (\frac{x ^{2}}{4}) + 42 = 0$

Forming the Quadratic

Multiply the entire equation by $4$ to remove the fraction:
$4 x - 2 x^{2} + 162 = 0$
Rearrange into standard quadratic form:
$2 x^{2} - 4 x - 162 = 0$

Simplifying the Equation

Divide the equation by $2$ for simpler coefficients:
$x^{2} - 22 x - 16 = 0$
This is a standard quadratic $a x^{2} + b x + c = 0$ with $a = 1, b = - 22, c = - 16$ .

Sum and Product of Roots

Let the roots be $x_{1}$ and $x_{2}$ , representing the x-coordinates of the intersection points.
Sum of roots: $x_{1} + x_{2} = - \frac{b}{a} = 22$
Product of roots: $x_{1} x_{2} = \frac{c}{a} = - 16$

Calculating $∣ x_{1} - x_{2} ∣$

We need the difference between the roots: $∣ x_{1} - x_{2} ∣ = (x_{1} + x_{2})^{2} - 4 x_{1} x_{2}$
Substitute the sum and product:
$∣ x_{1} - x_{2} ∣ = (22)^{2} - 4 (- 16)$
$∣ x_{1} - x_{2} ∣ = 8 + 64 = 72 = 62$

Calculating $∣ y_{1} - y_{2} ∣$

Pro-Tip: Use the linear equation instead of the quadratic for simpler calculation!
From the line: $y = \frac{x}{2} + 4$
$∣ y_{1} - y_{2} ∣ = ∣ (\frac{x _{1}}{2} + 4) - (\frac{x _{2}}{2} + 4) ∣$
$∣ y_{1} - y_{2} ∣ = \frac{∣ x _{1} - x _{2} ∣}{2} = \frac{6 2}{2} = 6$

The Distance Formula

The length of the chord $L$ is the distance between $(x_{1}, y_{1})$ and $(x_{2}, y_{2})$ .
$L = (x_{1} - x_{2})^{2} + (y_{1} - y_{2})^{2}$
Substitute the calculated differences:
$L = (62)^{2} + (6)^{2}$

Final Calculation

$L = 72 + 36$
$L = 108$
$L = 36 \times 3 = 63$
The length of the chord is $63$ .

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The Sigma Insight: Standard Equations of Parabola, Ellipse, and Hyperbola

Solution Diagram

Analyzing the Setup

To find the intersection points of the parabola

x^{2} = 4 y

and the line

x - 2 y + 42 = 0

, we must solve the system simultaneously.

From the parabola equation, we have

y = \frac{x ^{2}}{4}

. Substituting this into the line equation gives:

x - 2 (\frac{x ^{2}}{4}) + 42 = 0

The Master Equation

To simplify, we multiply the entire equation by

4

to clear the fraction:

4 x - 2 x^{2} + 162 = 0

Rearranging into standard quadratic form

2 x^{2} - 4 x - 162 = 0

and dividing by

2

, we obtain:

x^{2} - 22 x - 16 = 0

The Power of Roots

Let the roots of this quadratic be

x_{1}

and

x_{2}

. By Vieta's formulas, we identify the sum and product of the roots:

x_{1} + x_{2} = 22

x_{1} x_{2} = - 16

We calculate the horizontal distance

∣ x_{1} - x_{2} ∣

using the identity

∣ x_{1} - x_{2} ∣ = (x_{1} + x_{2})^{2} - 4 x_{1} x_{2}

∣ x_{1} - x_{2} ∣ = (22)^{2} - 4 (- 16) = 8 + 64 = 72 = 62

The Geometric Shortcut

Since the points lie on the line

y = \frac{x}{2} + 4

, the vertical difference

∣ y_{1} - y_{2} ∣

is related to the horizontal difference by the slope of the line:

∣ y_{1} - y_{2} ∣ = \frac{∣ x _{1} - x _{2} ∣}{2}

Substituting our known value for the horizontal difference:

∣ y_{1} - y_{2} ∣ = \frac{6 2}{2} = 6

Final Calculation

The length of the chord

L

is the hypotenuse of the right triangle formed by the horizontal and vertical differences:

L = (x_{1} - x_{2})^{2} + (y_{1} - y_{2})^{2}

Plugging in the calculated values:

L = (62)^{2} + (6)^{2} = 72 + 36 = 108

Simplifying the radical, we find the final length of the chord:

$L = 63$

Common Traps & FAQs

Why do we use the sum and product of roots instead of the quadratic formula?

The quadratic formula is a blunt instrument. Using the sum and product of roots is like using a scalpel—it's precise, elegant, and minimizes the chance of arithmetic errors, which is exactly what you need in a high-pressure exam.

Why did we use the line equation to find

∣ y_{1} - y_{2} ∣

instead of the parabola equation?

The line is a linear relationship. The parabola is a quadratic one. By using the line, you avoid squaring terms and dealing with higher-order complexities. Always look for the linear path!

What if the line was vertical? Would this method still work?

Great question! If the line were vertical, the slope would be undefined, and the equation would be

x = k

. In that case, you would just find the two y-values by plugging

x = k

into the parabola equation. The method adapts to the geometry!

Animated Solution for Mathematics - Conic Sections: The length of the chord of the parabola $x^{2} = 4 y$ having equation $x - 2 y + 42 = 0$ is :

Select Answer:

Visualized Solution

Visualizing the Geometry

Intersection Strategy

Substituting $y$

Forming the Quadratic

Simplifying the Equation

Sum and Product of Roots

Calculating $∣ x_{1} - x_{2} ∣$

Calculating $∣ y_{1} - y_{2} ∣$

The Distance Formula

Final Calculation

The Sigma Insight: Standard Equations of Parabola, Ellipse, and Hyperbola

Analyzing the Setup

The Master Equation

The Power of Roots

The Geometric Shortcut

Final Calculation

Similar Questions

If one end of a focal chord of the parabola, $y^{2} = 16 x$ is at $(1, 4)$ , then the length of this focal chord is

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Animated Solution for Mathematics - Conic Sections: .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The length of the chord of the parabola x2=4y having equation x−2​y+42​=0 is :

Select Answer:

Visualized Solution

Visualizing the Geometry

Intersection Strategy

Substituting .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } y

Forming the Quadratic

Simplifying the Equation

Sum and Product of Roots

Calculating .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } ∣x1​−x2​∣

Calculating .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } ∣y1​−y2​∣

The Distance Formula

Final Calculation

The Sigma Insight: Standard Equations of Parabola, Ellipse, and Hyperbola

Analyzing the Setup

The Master Equation

The Power of Roots

The Geometric Shortcut

Final Calculation

Similar Questions

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.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If the x-intercept of a focal chord of the parabola y2=8x+4y+4 is 3, then the length of this chord is equal to

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Animated Solution for Mathematics - Conic Sections: The length of the chord of the parabola $x^{2} = 4 y$ having equation $x - 2 y + 42 = 0$ is :

Substituting $y$

Calculating $∣ x_{1} - x_{2} ∣$

Calculating $∣ y_{1} - y_{2} ∣$

If one end of a focal chord of the parabola, $y^{2} = 16 x$ is at $(1, 4)$ , then the length of this focal chord is

If the x-intercept of a focal chord of the parabola $y^{2} = 8 x + 4 y + 4$ is 3, then the length of this chord is equal to

Let the length of the focal chord $P Q$ of the parabola $y^{2} = 12 x$ be 15 units. If the distance of $P Q$ from the origin is $p$ , then $10 p^{2}$ is equal to _______

Let $S$ be the focus of the parabola $y^{2} = 8 x$ and let $P Q$ be the common chord of the circle $x^{2} + y^{2} - 2 x - 4 y = 0$ and the given parabola. The area of the triangle $P QS$ is

Show that the locus of a point that divides a chord of slope 2 of the parabola $y^{2} = 4 x$ internally in the ratio $1 : 2$ is a parabola. Find the vertex of this parabola.